Basic theorems of analysis of variance with special reference to field experiments - Rayner, A. A. (The University of Edinburgh, 1947)
The statistical procedure of analysis of variance was invented by R.A. Fisher during his stay as statistician at Rothamsted Experimental Station. His first, more or less tentative, discussion of the theory was set forth in a paper published in 1923 (11), and this was quickly followed up by the more assured and much more complete exposition in his book "Statistical Methods for Research Workers" (12), which revolutionised previous ideas on the principles of scientific experiment. Little additional work was published on the sllbject until 1933, but since then many workers, among whom may be mentioned M.S. Bartlett, W.G. Cochran, J. Wishart, and above all F. Yates, the present chief statistician at Rothamsted, have developed the theory on the lines laid down by Fisher. impetus was given to this development especially by the publications of Yates and of Fisher himself (13), in which the new methods of factorial design, confounding, and covariance introduced at Rothamsted were first made more generally known. Fisher's theories met with spasmodic opposition from statisticians such as "Student ", Neyman, and others, but have triumphed over all opposition and today are the basis of almost all scientific experimental work amenable to statistical treatment. Nevertheless one would look in vain throughout the literature for any rigorous and at the same time reasonably simple mathematical treatment of the theory of analysis of variance. Fisher's own exposition is for the most part seemingly intuitive, being designed for the non- mathematical reader, as are for the most part the papers of Yates. Modern text -books such as Snedecor's "Statistical Methods" (28) present the methods without the theory behind them and appeal to the intuition of the reader. Where proofs are attempted, vital points are usually glossed over or assumed, as being beyond the scope of an elementary book. Among the very few British mathematical papers on analysis of variance are those of Irwin (15,16), but his treatment is complicated and unwieldy. Cochran (6) realised the advantages of matrix notation in a subject of this sort, and many of his theorems are equivalent to the lemmas of this thesis, but Cochran left the application of his method undeveloped. The present thesis constitutes an attempt to put forward a progressive mathematical theory of analysis of variance as applied to the various situations met with in agricultural research in particular, but the applications are, of course, perfectly general. Matrix notation has been used throughout to simplify a subject which would otherwise prove rather unwieldy for mathematical treatment. The basic theories are those of Fisher, Yates, etc., and are now so generally accepted as to require no special references. Acknowledgment by reference is therefore made only in the case of specific points where this has seemed necessary.
The basic theorems of analysis of variance with special reference to field experiments